Nzvimbo yeMutsetse Uchipesana neDenderedzwa
Denderedzwa chimiro chakakosha chejometri ine mashandisirwo akasiyana-siyana muzvikamu zvakasiyana-siyana, kubva pamasvomhu kusvika kufizikisi neinjiniya. Chinhu chakakosha pakudzidza madenderedzwa kunzwisisa nzvimbo yemitsara maererano nawo. Nzvimbo iyi inokosha mumamiriro ezvinhu akawanda, akadai sekugadzira geometric, kuongorora chimiro, uye kudzidza kwezviratidzo zvepfungwa nemasvomhu.
1. Tsanangudzo yeMitsetse neMadenderedzwa
Denderedzwa iboka remapoinzi ese ari mudenga ari chinhambwe chisingachinji kubva panzvimbo iri pakati. Mutsetse iboka remapoinzi anoumba mutsetse wakatwasuka usingaperi.
Pamasvomhu, denderedzwa rine pakati (h, k) uye radius r rinoratidzwa ne equation:
\[ (x – h)^2 + (y – k)^2 = r^2 \]
Mitsetse inogona kuratidzwa nenzira dzakasiyana-siyana. Chimiro chemutsetse uri muCartesian coordinate plane ndeichi:
\[ Ax + By + C = 0 \]
2. Nzvimbo yemutsetse maererano nedenderedzwa
Nzvimbo yemutsetse kana tichienzanisa nedenderedzwa inogona kuiswa muzvikamu zvitatu zvikuru:
1. Mutsetse weTangent
2. Mutsetse Wepachivande
3. Mitsetse Kunze kweDenderedzwa
Mutsetse weTangent kuenda kuDenderedzwa
Mutsetse unotenderedza denderedzwa mutsetse unopindirana nedenderedzwa panzvimbo imwe chete. Iyi nzvimbo inotenderedza inonzi nzvimbo inotenderedza denderedzwa. Pachishandiswa chimiro chemufananidzo, mutsetse unotenderedza denderedzwa kana uye chete kana:
\[d = r \]
Apo d iri chinhambwe kubva pakati pedenderedzwa kusvika pamutsetse, uye r iri dhayamita yedenderedzwa.
Kuti tizive daro kubva pakati pedenderedzwa (h, k) kusvika pamutsetse Ax + By + C = 0, tinoshandisa fomura iyi:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
Kana \( d = r \), mutsetse wacho wakabatana nedenderedzwa.
Denderedzwa Rinopindirana Nemitsetse
Mutsetse unoganhura denderedzwa panzvimbo mbiri dzakasiyana. Muchiitiko ichi, mutsetse wacho uri secant yedenderedzwa. Pamasvomhu, mutsetse unoganhura denderedzwa kana daro kubva pakati pedenderedzwa kuenda kumutsetse riri pasi pe radius yedenderedzwa:
\[ d < r \] Mutsetse Kunze kweDenderedzwa Mutsetse uri kunze kwedenderedzwa kana daro kubva pakati pedenderedzwa kuenda kumutsetse rakakura kupfuura radius yedenderedzwa: \[ d > r \]
3. Kuongororwa kweChinzvimbo cheMutsetse maererano neDenderedzwa neMienzaniso
Muenzaniso unotevera unoratidza kunzwisisa kwenzvimbo yemutsetse kana tichienzanisa nedenderedzwa.
Muenzaniso 1: Mutsetse weTangent kuenda kuDenderedzwa
Ngatitii tine denderedzwa rine pakati (3, 2) uye radius 5. Mubvunzo ndewokuti mutsetse \( x + 2y = 7 \) wakabatana nedenderedzwa here?
Danho rekutanga, tinowana daro kubva pakati pedenderedzwa kusvika pamutsetse.
\[ h = 3, \, k = 2, \, r = 5 \]
\[ d = \frac{|1\cdot3 + 2\cdot2 – 7|}{\sqrt{1^2 + 2^2}} = \frac{|3 + 4 – 7|}{\sqrt{5}} = \frac{0}{\sqrt{5}} = 0 \]
Sezvo \( d \neq r \), mutsetse haubati denderedzwa. Ngationgororei nekuverenga zvakare:
\[
d = \frac{|1\cdot3 + 2\cdot2 – 7|}{\sqrt{1^2 + 2^2}} = \frac{|3 + 4 – 7|}{\sqrt{5}} = \frac{0}{\sqrt{5}} = 0
\]
Zvinosuwisa, semuenzaniso, kana paine chikanganiso kana \( d \neq r \), tichaedza mutsetse \( x + 2y = 8 \)
Danho rekutanga, tinowana daro kubva pakati pedenderedzwa kusvika pamutsetse.
\[ h = 3, \, k = 2, \, r = 5 \]
\[ d = \frac{|1\cdot3 + 2\cdot2 – 8|}{\sqrt{1^2 + 2^2}} = \frac{|3 + 4 – 8|}{\sqrt{5}} = \frac{1}{\sqrt{5}} = 4,7 \] kana d
Sezvo \( d < r \), mutsetse haubati denderedzwa. Muenzaniso wechipiri: Mutsetse Unopatsanura Denderedzwa Iye zvino, tine denderedzwa rine pakati (0, 0) uye radius 3. Ngationei kana mutsetse y = x + 1 uchipatsanura denderedzwa.
Danho rekutanga, tinowana daro kubva pakati pedenderedzwa kuenda kumutsetse. \[ h = 0, \, k = 0, \, r = 3 \] \[ A = 1, \, B = -1, \, C = -1 \] \[ d = \frac{|1\cdot0 - 1\cdot0 - 1|}{\sqrt{1^2 + (-1)^2}} = \frac{|-1|}{\sqrt{2}} = \frac{1}{\sqrt{2}} \approx 0.71 \] Sezvo \( d < r \), mutsetse uyu unopindirana nedenderedzwa panzvimbo mbiri. 4. Mhedziso Nzvimbo yemutsetse une chekuita nedenderedzwa ipfungwa huru mu geometry inobatsira mumashandisirwo akasiyana-siyana. Nzvimbo iyi inogona kupatsanurwa zvichibva padaro kubva pakati pedenderedzwa kuenda kumutsetse. Kana daro rakaenzana neradius yedenderedzwa, mutsetse wacho unobatana nedenderedzwa. Kana daro riri pasi pe nharaunda yedenderedzwa, mutsetse unopindirana nedenderedzwa. Kana daro riri guru kupfuura nharaunda, mutsetse uri kunze kwedenderedzwa. Kunzwisisa nzvimbo yemutsetse maererano nedenderedzwa kunobatsira mukuongorora kwakasiyana-siyana kwejometri uye mamwe mashandisirwo anoshanda, kubva pakuronga nekugadzira mainjiniya kusvika pakutsvagisa kwesainzi kwakaoma. Nekunzwisisa kwakasimba kwenzvimbo iyi, nyanzvi kana muongorori anogona kugadzira nekuongorora maumbirwo nemazvo uye zvinobudirira.